A boat travels 12 km upstream and back in 1 hour 45 minutes. If the speed of the current is 3km/h throughout,find the speed of the boat in still water, giving your answer correct to 3 significant...

A boat travels 12 km upstream and back in 1 hour 45 minutes. If the speed of the current is 3km/h throughout,

find the speed of the boat in still water, giving your answer correct to 3 significant figures.

Expert Answers
kjcdb8er eNotes educator| Certified Educator

distance = velocity x time

Let v = speed of the boat in still water

s = speed of the current = 3 km/h

Traveling upstream, the boat goes 12 km against the current at speed v - s in time t1: 12 km = (v - s)*t1

Traveling downstream on the return trip, the boat goes 12 km with the current at speed v + s in time t2: 12 km = (v + s)*t2

The total time T of the trip, t1 + t2, is 1.75 hr.

So: t2 = 12 km/(v + s) and t1 + t2 = T, and t1 - t2 = T - 2*t2

12 km - 12 km = (v - s)*t1 - (v + s)*t2

0 = v*(t1 - t2) - s*(t1 + t2) = v*(T - 2t2) - sT = v*(T - 2*12 km/(v + s) ) - sT

Multiplying both sides by (v + s), we get

0 = vT(v + s) - (24 km)v - sT(v + s) = Tv^2 - (24 km)v -Ts^2

Plugging in s = 3 km/h and T = 1.75 h, we get

0 = 1.75v^2 - 24v - 15.75

using the quadratic formula,

v = (24 +/- sqrt(24^2 - 4*1.75*(-15.75)))/(2*1.75)

v = 14.3418 km/h or - 0.62754 km/h

The first of these is clearly the physical answer, since we assumed that v was positive in our definitions above.

So, to three significant digits,

v = 14.3 km/h